|
|
|
|
References
|
|
| |
| 1. |
K. B. Athreya, P. E. Ney, Branching processes, Die Grundlehren der mathematischen Wissenschaften, 196, Springer-Verlag, New York–Heidelberg, 1972 |
| 2. |
E. D. Andjel, “Invariant measures for the zero range processes”, Ann. Probab., 10:3 (1982), 525–547 |
| 3. |
V. Ya. Basis, “Infinite-dimensional Markov processes with almost local interaction of the components”, Teor. Verojatnost. i Primenen., 21:4 (1976), 727–740 (Russian) ; English translation: Theory Probability and its Applications, 21:4 (1977), 706–720 |
| 4. |
V. Ya. Basis, “Stationarity and ergodicity of Markov interacting processes”, Multicomponent random systems, Adv. Probab. Related Topics, 6, Dekker, New York, 1980, 37–58 |
| 5. |
V. Bezborodov, Spatial birth-and-death Markov processes, Ph.D. thesis, Universität Bielefeld, September 2014 |
| 6. |
V. Bezborodov, Spatial birth-and-death Markov dynamics of finite particle systems, 1507, arXiv: 1507.05804 [math.PR] |
| 7. |
M. Balázs, G. Farkas, P. Kovács, A. Rákos, “Random walk of second class particles in product shock measures”, J. Stat. Phys., 139:2 (2010), 252–279 |
| 8. |
B. M. Bolker, S. W. Pacala, “Using moment equations to understand stochastically driven spatial pattern formation in ecological systems”, Theoretical Population Biology, 52:3 (1997), 179–197 |
| 9. |
B. M. Bolker, S. W. Pacala, “Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal”, The American Naturalist, 153:6 (1999), 575–602 |
| 10. |
U. Dieckmann, R. Law, “Relaxation projections and the method of moments”, The Geometry of Ecological Interactions. Simplifying Spatial Complexity, eds. U. Dieckmann, R. Law, J. A. J. Metz, Cambridge University Press, 2000, 412–455 |
| 11. |
A. de La Fortelle, “Yule process sample path asymptotics”, Electron. Comm. Probab., 11 (2006), 193–199 |
| 12. |
S. N. Ethier, T. G. Kurtz, Markov processes. Characterization and convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986 |
| 13. |
A. M. Etheridge, “Survival and extinction in a locally regulated population”, Ann. Appl. Probab., 14:1 (2004), 188–214 |
| 14. |
D. Finkelshtein, Y. Kondratiev, O. Kutoviy, “Individual based model with competition in spatial ecology”, SIAM J. Math. Anal., 41:1 (2009), 297–317 |
| 15. |
D. Finkelshtein, Y. Kondratiev, O. Kutoviy, “Semigroup approach to birth-and-death stochastic dynamics in continuum”, J. Funct. Anal., 262:3 (2012), 1274–1308 |
| 16. |
D. Finkelshtein, Y. Kondratiev, O. Kutoviy, “An operator approach to Vlasov scaling for some models of spatial ecology”, Methods Funct. Anal. Topology, 19:2 (2013), 108–126 |
| 17. |
D. Finkelshtein, Y. Kondratiev, O. Kutoviy, E. Zhizhina, “On an aggregation in birth-and-death stochastic dynamics”, Nonlinearity, 27:6 (2014), 1105–1133 |
| 18. |
N. Fournier, S. Méléard, “A microscopic probabilistic description of a locally regulated population and macroscopic approximations”, Ann. Appl. Probab., 14:4 (2004), 1880–1919 |
| 19. |
N. L. Garcia, T. G. Kurtz, “Spatial birth and death processes as solutions of stochastic equations”, ALEA Lat. Am. J. Probab. Math. Stat., 1 (2006), 281–303 |
| 20. |
R. A. Holley, D. W. Stroock, “A martingale approach to infinite systems of interacting processes”, Ann. Probability, 4:2 (1976), 195–228 |
| 21. |
N. Ikeda, S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam–New York; Kodansha, Ltd., Tokyo, 1981 |
| 22. |
O. Kallenberg, Foundations of modern probability, Probability and its Applications (New York), second ed., Springer-Verlag, New York, 2002 |
| 23. |
Y. G. Kondratiev, T. Kuna, “Harmonic analysis on configuration space. I. General theory”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5:2 (2002), 201–233 |
| 24. |
Y. G. Kondratiev, O. Kutoviy, R. Minlos, “Ergodicity of non-equilibrium Glauber dynamics in continuum”, J. Funct. Anal., 258:9 (2010), 3097–3116 |
| 25. |
Y. G. Kondratiev, O. V. Kutoviy, E. Zhizhina, “Nonequilibrium Glauber-type dynamics in continuum”, J. Math. Phys., 47:11 (2006), 113501, 17 pp. |
| 26. |
T. G. Kurtz, P. E. Protter, “Weak convergence of stochastic integrals and differential equations. II. Infinite-dimensional case”, Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), Lecture Notes in Math., 1627, Springer, Berlin, 1996, 197–285 |
| 27. |
T. G. Kurtz, “The Yamada–Watanabe–Engelbert theorem for general stochastic equations and inequalities”, Electron. J. Probab., 12 (2007), 951–965 |
| 28. |
T. M. Liggett, Interacting particle systems, Grundlehren der Mathematischen Wissenschaften, 276, Springer-Verlag, New York, 1985 |
| 29. |
T. M. Liggett, F. Spitzer, “Ergodic theorems for coupled random walks and other systems with locally interacting components”, Z. Wahrsch. Verw. Gebiete, 56:4 (1981), 443–468 |
| 30. |
D. J. Murrell, U. Dieckmann, R. Law, “On moment closures for population dynamics in continuous space”, J. Theor. Biol., 229 (2004), 421–432 |
| 31. |
O. Ovaskainen, D. Finkelshtein, O. Kutoviy, S. Cornell, B. Bolker, Y. Kondratiev, “A general mathematical framework for the analysis of spatiotemporal point processes”, Theoretical Ecology, 7:1 (2014), 101–113 |
| 32. |
M. D. Penrose, “Existence and spatial limit theorems for lattice and continuum particle systems”, Probab. Surv., 5 (2008), 1–36 |